A Novel Implementation of the Matrix Element Method at Next-to-Leading Order for the Measurement of the Higgs Self-Coupling $λ_{3H}$

The determination of the Higgs boson trilinear self-coupling $λ_{3H}$ is a key goal of the LHC physics programme. Its precise measurement will provide unique insight into the scalar potential and the mechanism of electroweak symmetry breaking. Higgs boson pair production in the ${gg}\to{HH}$ process, and particularly in the ${HH}\to{b}\bar{b}γγ$ final state, offers direct sensitivity to $λ_{3H}$. We present the first implementation of the Matrix Element Method at Next-to-Leading Order (MEM@NLO) for this process, which is publicly available. The MEM is a statistically optimal approach that maximises information extraction from collision events. Extending it to NLO represents a major methodological challenge, which we address with a new formalism integrated into the MoMEMta framework. Results with simulated pseudo-experiments demonstrate, in a proof-of-principle study, the strong discriminating power of the method and its ability to extract the coupling modifier $κ_λ$=$λ_{3H}$/$λ_{3H}^{SM}$ with high precision.

💡 Research Summary

The paper presents the first implementation of the Matrix Element Method (MEM) at next‑to‑leading order (NLO) for the gluon‑fusion Higgs‑pair production process gg→HH→b b̄ γγ, a channel that directly probes the Higgs trilinear self‑coupling λ₃ᴴ. Precise determination of λ₃ᴴ, expressed through the coupling modifier κλ = λ₃ᴴ/λ₃ᴴᴿᴱᴰ, is a central goal of the LHC physics programme because deviations from the Standard Model value would signal new dynamics in the scalar sector and could be linked to the matter‑antimatter asymmetry. Current experimental limits from ATLAS and CMS still allow κλ to vary between roughly –1.4 and 7, reflecting the difficulty of measuring a process with a tiny cross‑section and limited statistics.

To overcome the statistical limitations, the authors exploit the MEM, which evaluates the probability that a measured event originates from a given hypothesis by integrating the partonic matrix element over all parton‑level configurations compatible with the reconstructed observables. Traditional MEM applications have been restricted to leading‑order (LO) matrix elements, ignoring additional QCD radiation that is inevitably present in realistic LHC events. This omission leads to a mismatch between the theoretical description and the data, reducing the discriminating power of the method.

The core contribution of the work is a full NLO extension of the MEM, dubbed MEM@NLO, built on the modular MoMEMta framework. The authors first modify the POWHEG‑BOX‑V2 implementation of gg→HH (the “ggHH” sub‑repository) to expose Born, virtual, and real‑emission contributions at arbitrary phase‑space points. Because the real‑emission term introduces an extra parton, the dimensionality of the integration increases and the phase‑space mapping becomes non‑trivial. To address this, a new integration block, called Block N, is introduced. Block N simultaneously eliminates the two initial‑state Bjorken‑x variables and the transverse components of the unresolved radiated parton by using four‑momentum conservation, while keeping the longitudinal component of the radiated momentum as a free integration variable. The resulting Jacobian is a simple factor of ½, dramatically simplifying the numerical integration.

Infrared safety is ensured by using the infrared‑finite matrix elements supplied by the modified POWHEG‑BOX, and by imposing a minimal transverse‑momentum cut on the unresolved parton together with a dedicated collinear‑treatment module that reshapes configurations where the extra parton becomes collinear with an incoming gluon or with a Higgs decay product. These technical regulators do not alter the underlying NLO prediction but prevent inefficient sampling of singular regions.

The authors validate the implementation through several cross‑checks. They compare the Born and real‑emission matrix elements obtained from POWHEG‑BOX‑V2 with those from MadGraph5_aMC@NLO across representative phase‑space points, finding excellent agreement. They then evaluate the discriminating power of MEM@NLO using receiver‑operating‑characteristic (ROC) curves built from the likelihood ratio between the signal hypothesis (gg→HH) and the dominant background (ttH). The ROC curve for MEM@NLO lies significantly closer to the ideal top‑right corner than the LO counterpart, demonstrating that accounting for NLO radiation restores and even enhances the method’s sensitivity.

Finally, the paper shows how MEM@NLO can be employed to extract κλ. For each event, the per‑process likelihood Lₚ(κλ|xᵢ) is computed using the NLO matrix elements and transfer functions that model detector response. The event‑level likelihood is then formed by weighting the process‑level contributions according to their expected fractions. By multiplying the event likelihoods over the full dataset, a global likelihood L(κλ) is obtained. Scanning this likelihood as a function of κλ yields a precise measurement: in pseudo‑experiments the method recovers the Standard Model value κλ = 1 with a substantially reduced uncertainty compared to LO‑based analyses.

In summary, this work delivers a publicly available, fully validated MEM@NLO implementation for Higgs‑pair production in the b b̄ γγ final state. By integrating NLO QCD effects directly into the MEM, it achieves superior background discrimination and enables high‑precision extraction of the Higgs self‑coupling. The approach is readily extensible to other rare processes and will be especially valuable in the high‑luminosity LHC era, where maximizing the information extracted from each event is crucial for uncovering subtle signs of physics beyond the Standard Model.